## Abstract and Applied Analysis

### The Lie Group in Infinite Dimension

#### Abstract

A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local, ${C}^{\infty}$ smooth) action of a Lie group on infinite-dimensional space (a manifold modelled on $\mathbb{R}^{\infty}$) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

#### Article information

Source
Abstr. Appl. Anal., Volume 2011, Number 1 (2011), Article ID 919538, 35 pages.

Dates
First available in Project Euclid: 12 August 2011

https://projecteuclid.org/euclid.aaa/1313171392

Digital Object Identifier
doi:10.1155/2011/919538

Mathematical Reviews number (MathSciNet)
MR2771243

Zentralblatt MATH identifier
1223.22018

#### Citation

Tryhuk, V.; Chrastinová, V.; Dlouhý, O. The Lie Group in Infinite Dimension. Abstr. Appl. Anal. 2011 (2011), no. 1, Article ID 919538, 35 pages. doi:10.1155/2011/919538. https://projecteuclid.org/euclid.aaa/1313171392